0000000000005485
AUTHOR
Ville Suomala
showing 12 related works from this author
Conical upper density theorems and porosity of measures
2008
Abstract We study how measures with finite lower density are distributed around ( n − m ) -planes in small balls in R n . We also discuss relations between conical upper density theorems and porosity. Our results may be applied to a large collection of Hausdorff and packing type measures.
Existence of doubling measures via generalised nested cubes
2012
Working on doubling metric spaces, we construct generalised dyadic cubes adapting ultrametric structure. If the space is complete, then the existence of such cubes and the mass distribution principle lead into a simple proof for the existence of doubling measures. As an application, we show that for each $\epsilon>0$ there is a doubling measure having full measure on a set of packing dimension at most $\epsilon$.
Nonsymmetric conical upper density and $k$-porosity
2017
We study how the Hausdorff measure is distributed in nonsymmetric narrow cones in R n \mathbb {R}^n . As an application, we find an upper bound close to n − k n-k for the Hausdorff dimension of sets with large k k -porosity. With k k -porous sets we mean sets which have holes in k k different directions on every small scale.
Thin and fat sets for doubling measures in metric spaces
2011
We consider sets in uniformly perfect metric spaces which are null for every doubling measure of the space or which have positive measure for all doubling measures. These sets are called thin and fat, respectively. In our main results, we give sufficient conditions for certain cut-out sets being thin or fat.
On the conical density properties of measures on $\mathbb{R}^n$
2005
We compare conical density properties and spherical density properties for general Borel measures on $\mathbb{R}^n$ . As a consequence, we obtain results for packing and Hausdorff measures $\mathcal{P}_h$ and $\mathcal{H}_h$ provided that the gauge function $h$ satisfies certain conditions. One consequence of our general results is the following: let $m, n\,{\in}\,\mathbb{N}, 0\,{\lt}\,s\,{\lt}\,m\,{\leq}\,n$ , $0\,{\lt}\,\eta\,{\lt}\,1$ , and suppose that $V$ is an $m$ -dimensional linear subspace of $\mathbb{R}^n$ . Let $\mu$ be either the $s$ -dimensional Hausdorff measure or the $s$ -dimensional packing measure restricted to a set $A$ with $\mu(A)\,{\lt}\,\infty$ . Then for $\mu$ -almos…
Dimensions of random affine code tree fractals
2014
We calculate the almost sure Hausdorff dimension for a general class of random affine planar code tree fractals. The set of probability measures describing the randomness includes natural measures in random $V$-variable and homogeneous Markov constructions.
Which measures are projections of purely unrectifiable one-dimensional Hausdorff measures
2008
We give a necessary and sufficient condition for a measure p, on the real line to be an orthogonal projection of XAl for some purely 1-unrectifiable planar set A.
Local multifractal analysis in metric spaces
2013
We study the local dimensions and local multifractal properties of measures on doubling metric spaces. Our aim is twofold. On one hand, we show that there are plenty of multifractal type measures in all metric spaces which satisfy only mild regularity conditions. On the other hand, we consider a local spectrum that can be used to gain finer information on the local behaviour of measures than its global counterpart.
Local conical dimensions for measures
2012
AbstractWe study the decay of μ(B(x,r)∩C)/μ(B(x,r)) asr↓ 0 for different kinds of measures μ on ℝnand various conesCaroundx. As an application, we provide sufficient conditions that imply that the local dimensions can be calculated via cones almost everywhere.
Random cutout sets with spatially inhomogeneous intensities
2015
We study the Hausdorff dimension of Poissonian cutout sets defined via inhomogeneous intensity measures on Ahlfors-regular metric spaces. We obtain formulas for the Hausdorff dimension of such cutouts in self-similar and self-conformal spaces using the multifractal decomposition of the average densities for the natural measures.
Tangential behavior of functions and conical densities of Hausdorff measures.
2005
We construct a $C^1$-function $f\colon [0,1]\to \mathbb{R}$ such that for almost all $x\in(0,1)$, there is $r>0$ for which $f(y)>f(x)+f'(x)(y-x)$ when $y\in(x,x+r)$ and $f(y)< f(x)+f'(x)(y-x)$ when $y\in(x-r,x)$. The existence of such functions is related to a problem concerning conical density properties of Hausdorff measures on $\mathbb{R}^n$. We also discuss the tangential behavior of typical $C^1$-functions, using an improvement of Jarnik's theorem on essential derived numbers
Local dimensions in Moran constructions
2015
We study the dimensional properties of Moran sets and Moran measures in doubling metric spaces. In particular, we consider local dimensions and $L^q$-dimensions. We generalize and extend several existing results in this area.